3.5.26 \(\int \frac {a+b \log (c x^n)}{x (d+e x^r)^3} \, dx\) [426]

Optimal. Leaf size=169 \[ -\frac {b n}{2 d^2 r^2 \left (d+e x^r\right )}-\frac {b n \log (x)}{2 d^3 r}+\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}+\frac {3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^2} \]

[Out]

-1/2*b*n/d^2/r^2/(d+e*x^r)-1/2*b*n*ln(x)/d^3/r+1/2*(a+b*ln(c*x^n))/d/r/(d+e*x^r)^2-e*x^r*(a+b*ln(c*x^n))/d^3/r
/(d+e*x^r)-(a+b*ln(c*x^n))*ln(1+d/e/(x^r))/d^3/r+3/2*b*n*ln(d+e*x^r)/d^3/r^2+b*n*polylog(2,-d/e/(x^r))/d^3/r^2

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Rubi [A]
time = 0.29, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2391, 2379, 2438, 2373, 266, 2376, 272, 46} \begin {gather*} \frac {b n \text {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^3 r^2}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}+\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}+\frac {3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}-\frac {b n \log (x)}{2 d^3 r}-\frac {b n}{2 d^2 r^2 \left (d+e x^r\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(x*(d + e*x^r)^3),x]

[Out]

-1/2*(b*n)/(d^2*r^2*(d + e*x^r)) - (b*n*Log[x])/(2*d^3*r) + (a + b*Log[c*x^n])/(2*d*r*(d + e*x^r)^2) - (e*x^r*
(a + b*Log[c*x^n]))/(d^3*r*(d + e*x^r)) - ((a + b*Log[c*x^n])*Log[1 + d/(e*x^r)])/(d^3*r) + (3*b*n*Log[d + e*x
^r])/(2*d^3*r^2) + (b*n*PolyLog[2, -(d/(e*x^r))])/(d^3*r^2)

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2373

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Dist[b*(n/(d*(m + 1))), Int[(f*x)^
m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[
m, -1]

Rule 2376

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :
> Simp[f^m*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^p/(e*r*(q + 1))), x] - Dist[b*f^m*n*(p/(e*r*(q + 1))), Int[
(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[
m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2391

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_))/(x_), x_Symbol] :> Dist[1/d,
Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[x^(r - 1)*(d + e*x^r)^q*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0] && ILtQ[q, -1]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^2} \, dx}{d}-\frac {e \int \frac {x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^3} \, dx}{d}\\ &=\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx}{d^2}-\frac {e \int \frac {x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx}{d^2}-\frac {(b n) \int \frac {1}{x \left (d+e x^r\right )^2} \, dx}{2 d r}\\ &=\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}-\frac {(b n) \text {Subst}\left (\int \frac {1}{x (d+e x)^2} \, dx,x,x^r\right )}{2 d r^2}+\frac {(b n) \int \frac {\log \left (1+\frac {d x^{-r}}{e}\right )}{x} \, dx}{d^3 r}+\frac {(b e n) \int \frac {x^{-1+r}}{d+e x^r} \, dx}{d^3 r}\\ &=\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}+\frac {b n \log \left (d+e x^r\right )}{d^3 r^2}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^2}-\frac {(b n) \text {Subst}\left (\int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx,x,x^r\right )}{2 d r^2}\\ &=-\frac {b n}{2 d^2 r^2 \left (d+e x^r\right )}-\frac {b n \log (x)}{2 d^3 r}+\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}+\frac {3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^2}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 170, normalized size = 1.01 \begin {gather*} \frac {\frac {d^2 r \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2}+\frac {d \left (-b n+2 a r+2 b r \log \left (c x^n\right )\right )}{d+e x^r}+3 b n \log \left (d-d x^r\right )-2 a r \log \left (d-d x^r\right )+2 b r \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+2 b n \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\text {Li}_2\left (1+\frac {e x^r}{d}\right )\right )}{2 d^3 r^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Log[c*x^n])/(x*(d + e*x^r)^3),x]

[Out]

((d^2*r*(a + b*Log[c*x^n]))/(d + e*x^r)^2 + (d*(-(b*n) + 2*a*r + 2*b*r*Log[c*x^n]))/(d + e*x^r) + 3*b*n*Log[d
- d*x^r] - 2*a*r*Log[d - d*x^r] + 2*b*r*(n*Log[x] - Log[c*x^n])*Log[d - d*x^r] + 2*b*n*((r^2*Log[x]^2)/2 + (-(
r*Log[x]) + Log[-((e*x^r)/d)])*Log[d + e*x^r] + PolyLog[2, 1 + (e*x^r)/d]))/(2*d^3*r^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.17, size = 1012, normalized size = 5.99

method	result	size
risch	\(\text {Expression too large to display}\)	\(1012\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x/(d+e*x^r)^3,x,method=_RETURNVERBOSE)

[Out]

1/2*I/r*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^3*ln(d+e*x^r)-1/2*I/r*b*Pi*csgn(I*c*x^n)^3/d^2/(d+e*x^r)+b/
r/d^3*ln(d+e*x^r)*n*ln(x)-b/r/d^2/(d+e*x^r)*n*ln(x)-b/r*n/d^3*ln(x)*ln((d+e*x^r)/d)-1/2*b/r/d/(d+e*x^r)^2*n*ln
(x)-b/r/d^3*ln(x^r)*n*ln(x)-1/4*I/r*b*Pi*csgn(I*c*x^n)^3/d/(d+e*x^r)^2+1/r*b*ln(c)/d^2/(d+e*x^r)+1/2/r*b*ln(c)
/d/(d+e*x^r)^2+1/r*b*ln(c)/d^3*ln(x^r)-b/r/d^3*ln(d+e*x^r)*ln(x^n)-1/2*I/r*b*Pi*csgn(I*c*x^n)^3/d^3*ln(x^r)-1/
2*b*n/d^2/r^2/(d+e*x^r)-1/2*b/r*n*e^2/d^3*ln(x)*(x^r)^2/(d+e*x^r)^2-b/r*n*e/d^2*ln(x)*x^r/(d+e*x^r)^2+1/2*I/r*
b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3*ln(x^r)+1/4*I/r*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d/(d+e*x^r)^2+a/r/d^3*ln
(x^r)+a/r/d^2/(d+e*x^r)+1/2*a/r/d/(d+e*x^r)^2+1/4*I/r*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d/(d+e*x^r)^2-a/r/d^3*ln(
d+e*x^r)-1/2*I/r*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3*ln(d+e*x^r)-b/r*n*e/d^3*ln(x)*x^r/(d+e*x^r)-1/2*I/r*b*Pi
*csgn(I*c)*csgn(I*c*x^n)^2/d^3*ln(d+e*x^r)+1/2*I/r*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^2/(d+e*x^r)+1/2*I/r*b*Pi*c
sgn(I*c*x^n)^3/d^3*ln(d+e*x^r)+1/2*I/r*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^3*ln(x^r)+1/2*I/r*b*Pi*csgn(I*x^n)*csg
n(I*c*x^n)^2/d^2/(d+e*x^r)+1/2*b*n/d^3*ln(x)^2-1/r*b*ln(c)/d^3*ln(d+e*x^r)+b/r/d^2/(d+e*x^r)*ln(x^n)+1/2*b/r/d
/(d+e*x^r)^2*ln(x^n)+b/r/d^3*ln(x^r)*ln(x^n)-b/r^2*n/d^3*dilog((d+e*x^r)/d)-1/2*I/r*b*Pi*csgn(I*c)*csgn(I*x^n)
*csgn(I*c*x^n)/d^2/(d+e*x^r)-1/2*I/r*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^3*ln(x^r)-1/4*I/r*b*Pi*csgn(I*
c)*csgn(I*x^n)*csgn(I*c*x^n)/d/(d+e*x^r)^2+3/2*b*n*ln(d+e*x^r)/d^3/r^2

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x/(d+e*x^r)^3,x, algorithm="maxima")

[Out]

1/2*a*((2*e*x^r + 3*d)/(d^2*e^2*r*x^(2*r) + 2*d^3*e*r*x^r + d^4*r) + 2*log(x)/d^3 - 2*log((e*x^r + d)/e)/(d^3*
r)) + b*integrate((log(c) + log(x^n))/(e^3*x*x^(3*r) + 3*d*e^2*x*x^(2*r) + 3*d^2*e*x*x^r + d^3*x), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (163) = 326\).
time = 0.35, size = 411, normalized size = 2.43 \begin {gather*} \frac {b d^{2} n r^{2} \log \left (x\right )^{2} + 3 \, b d^{2} r \log \left (c\right ) - b d^{2} n + 3 \, a d^{2} r + {\left (b n r^{2} e^{2} \log \left (x\right )^{2} + {\left (2 \, b r^{2} e^{2} \log \left (c\right ) - {\left (3 \, b n r - 2 \, a r^{2}\right )} e^{2}\right )} \log \left (x\right )\right )} x^{2 \, r} + {\left (2 \, b d n r^{2} e \log \left (x\right )^{2} + 2 \, b d r e \log \left (c\right ) - {\left (b d n - 2 \, a d r\right )} e + 4 \, {\left (b d r^{2} e \log \left (c\right ) - {\left (b d n r - a d r^{2}\right )} e\right )} \log \left (x\right )\right )} x^{r} - 2 \, {\left (2 \, b d n x^{r} e + b d^{2} n + b n x^{2 \, r} e^{2}\right )} {\rm Li}_2\left (-\frac {x^{r} e + d}{d} + 1\right ) - {\left (2 \, b d^{2} r \log \left (c\right ) - 3 \, b d^{2} n + 2 \, a d^{2} r + {\left (2 \, b r e^{2} \log \left (c\right ) - {\left (3 \, b n - 2 \, a r\right )} e^{2}\right )} x^{2 \, r} + 2 \, {\left (2 \, b d r e \log \left (c\right ) - {\left (3 \, b d n - 2 \, a d r\right )} e\right )} x^{r}\right )} \log \left (x^{r} e + d\right ) + 2 \, {\left (b d^{2} r^{2} \log \left (c\right ) + a d^{2} r^{2}\right )} \log \left (x\right ) - 2 \, {\left (2 \, b d n r x^{r} e \log \left (x\right ) + b d^{2} n r \log \left (x\right ) + b n r x^{2 \, r} e^{2} \log \left (x\right )\right )} \log \left (\frac {x^{r} e + d}{d}\right )}{2 \, {\left (2 \, d^{4} r^{2} x^{r} e + d^{5} r^{2} + d^{3} r^{2} x^{2 \, r} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x/(d+e*x^r)^3,x, algorithm="fricas")

[Out]

1/2*(b*d^2*n*r^2*log(x)^2 + 3*b*d^2*r*log(c) - b*d^2*n + 3*a*d^2*r + (b*n*r^2*e^2*log(x)^2 + (2*b*r^2*e^2*log(
c) - (3*b*n*r - 2*a*r^2)*e^2)*log(x))*x^(2*r) + (2*b*d*n*r^2*e*log(x)^2 + 2*b*d*r*e*log(c) - (b*d*n - 2*a*d*r)
*e + 4*(b*d*r^2*e*log(c) - (b*d*n*r - a*d*r^2)*e)*log(x))*x^r - 2*(2*b*d*n*x^r*e + b*d^2*n + b*n*x^(2*r)*e^2)*
dilog(-(x^r*e + d)/d + 1) - (2*b*d^2*r*log(c) - 3*b*d^2*n + 2*a*d^2*r + (2*b*r*e^2*log(c) - (3*b*n - 2*a*r)*e^
2)*x^(2*r) + 2*(2*b*d*r*e*log(c) - (3*b*d*n - 2*a*d*r)*e)*x^r)*log(x^r*e + d) + 2*(b*d^2*r^2*log(c) + a*d^2*r^
2)*log(x) - 2*(2*b*d*n*r*x^r*e*log(x) + b*d^2*n*r*log(x) + b*n*r*x^(2*r)*e^2*log(x))*log((x^r*e + d)/d))/(2*d^
4*r^2*x^r*e + d^5*r^2 + d^3*r^2*x^(2*r)*e^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x/(d+e*x**r)**3,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x/(d+e*x^r)^3,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((x^r*e + d)^3*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x^r\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(x*(d + e*x^r)^3),x)

[Out]

int((a + b*log(c*x^n))/(x*(d + e*x^r)^3), x)

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